Hard-particle packings have served as useful starting points to study the structure of diverse systems such as liquids, living cells, granular media, glasses, and amorphous solids, Howard Reiss has played a major role in helping to illuminate our understanding of hard-particle systems, which still offer scientists many interesting conundrums. Jammed configurations of hard particles are of great fundamental and practical interest. What one precisely means by a "jammed" configuration is quite subtle and considerable ambiguity remains in the literature on this question. We will show that there is a multiplicity of generation, selection, and classification procedures for jammed configurations of identical d-dimensional spheres. We categorize common ordered lattices according to our definitions and discuss implications for random disk and sphere packings. We also show how the concept of rigidity percolation (which has been used to understand the mechanical properties of network glasses) can be generalized to further characterize hard-sphere packings.